Noisy integrate-and-fire equation: continuation after blow-up

TL;DR

This paper develops a method to continue solutions of the integrate-and-fire equation after blow-up using random discharge models and a change of time, revealing the post-blow-up dynamics and blow-up phenomena.

Contribution

It introduces a novel approach combining random discharge models and a time change to extend solutions beyond blow-up in the integrate-and-fire equation.

Findings

Global solutions are recovered in the limit as epsilon approaches zero.

The approach describes the dynamics after blow-up using a singular measure.

New estimates handle nonlinear terms effectively.

Abstract

The integrate and fire equation is a classical model for neural assemblies which can exhibit finite time blow-up. A major open problem is to understand how to continue solutions after blow-up. Here we study an approach based on random discharge models and a change of time which generates a classical global solution to the expense of a strong absorption rate 1/$\epsilon$. We prove that in the limit $\epsilon$ $\rightarrow$ 0 + , a global solution is recovered where the integrate and fire equation is reformulated with a singular measure. This describes the dynamics after blow-up and also gives information on the blow-up phenomena itself.The major difficulty is to handle nonlinear terms. To circumvent it, we establish two new estimates, a kind of equi-integrability of the discharge measure and a L 2 estimate of the density. The use of the new timescale turns out to be fundamental for those estimates.